The diogonals AC and BD of a parallelogram ABCD bisect each other at O. A line segment XY through O has its end-points on the pposite sides AB and CD. Is XY also bisected at O.
Given : ABCD is a parallelogram. Diagonals AB and BD intersect at O. XY passes through O and intersects AB and CD in X and Y respectively.
To Prove : OX = OY
We know that, the diagonals of the parallelogram bisect each other.
∴ OA = OC
In ΔAOX and ΔCOY,
∠OAX = ∠OCY (Pair of alternate angles)
OA = OC
∠AOX= ∠COY (Vertically opposite angles)
∴ΔAOX ΔCOY (ASA congruence Criterion)
⇒ OX = OY (CPCT)
Thus, XY is bisected at O.